Geodesic
Conditions of commutativity of transfer matrices on a~multidimensional lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 52 (1982) no. 1, pp. 105-113

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It is shown that the transfer matrices for vertex models on a three-dimensional lattice commute if the statistical weights of the vertices satisfy a certain algebraic system of equations, which is called the “tetrahedron equation”. A multidimensional generalization of this proposition is also obtained, and it leads to "equations of $d$-simpliees". 
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     author = {V. V. Bazhanov and Yu. G. Stroganov},
     title = {Conditions of commutativity of transfer matrices on a~multidimensional lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     number = {1},
     year = {1982},
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V. V. Bazhanov; Yu. G. Stroganov. Conditions of commutativity of transfer matrices on a~multidimensional lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 52 (1982) no. 1, pp. 105-113. http://geodesic.mathdoc.fr/item/TMF_1982_52_1_a9/